Division by Zero: Indeterminate or Undefined?Date: 02/23/2002 at 16:21:24 From: Brant Langer Gurganus Subject: My theories on Zero In a previous response on a related topic, you said that division is defined in terms of reverse multiplication, and multiplication is defined in terms of repeated addition. Therefore, division is defined in terms of repeated subtraction. If so, then division is a count of how many times a number may be subtracted from another number until the original number equals zero. Case 1: 0/X; X is not 0 Operation: Count: 0 0 (it's already at 0 so 0/x=0) 0-X = -X (past 0) Case 2: X/0; X is not 0 Operation: Count: X 0 X-0 = X 1 X-0 = X 2 . . . . . . X-0 = X infinity (not undefined) Case 3: 0/0 Operation: Count: 0 0 0-0 = 0 1 . . . . . . 0-0 = 0 infinity (0/0 can have all numbers as solutions) Based on my cases above: 1. 0/x = 0 2. x/0 = infinity 3. 0/0 = one or more elements of the set of all numbers (depending on context) Can you comment on these conclusions? Date: 02/23/2002 at 17:41:35 From: Doctor Rick Subject: Re: My theories on Zero Hi, Brant. Your analysis seems like a reasonable way to describe and explain the results of dividing by zero. The only problem I have is that your use of the word "infinity" for both "infinitely large (greater than any number)" (what we call "undefined") and "can have any number as a solution" (what we call "indeterminate") is inconsistent and therefore confusing. See our explanations here: Dividing by 0 - Dr. Math FAQ http://mathforum.org/dr.math/faq/faq.divideby0.html The indeterminate nature of 0/0 http://mathforum.org/dr.math/problems/rob.12.21.00.html Defining 0/0 http://mathforum.org/dr.math/problems/howard.01.29.01.html - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 02/23/2002 at 19:04:08 From: Brant Langer Gurganus Subject: My theories on Zero What I meant was that since no matter how many times 0 is subtracted from 0, it will always be 0; so (expanding beyond integer division) all numbers can be a solution to 0/0. The "one or more solutions" depends on the context of the problem. Thanks for the quick reply. Date: 02/23/2002 at 21:08:29 From: Doctor Rick Subject: Re: My theories on Zero Hi, Brant. I know what you meant by "one or more solutions," and you're right. I hope you read the archive items I listed, which clarify how this works out in practice. As I put it, "indeterminate" means that there may be a definite answer to the problem you are working on, but you'll have to back up and find another way to discover it; dividing zero by zero cannot tell you the answer. It's sort of a "road closed" sign. A similar statement applies to "undefined." Rather than say that it takes an infinite number of subtractions of zero to get x down to 0, we have to say that NO number of subtractions of zero will get x down to 0. The answer to x/0 is not a number. That's another good way to say "undefined," and some computer languages call it this. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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